Let a problem instance be given as $(\phi(x_1,x_2,\dots, x_J),M)$ where $\phi$ is a diophantine equation, $J\leq 9$, and $M$ is a natural number. The decision problem is whether or not a given instance has a solution in natural numbers such that $\sum_{j=1}^J x_j \leq M$. With no upper bound M, the problem is undecidable (if I have the literature correct). With the bound, what is the computational complexity? If the equation does have such a solution, then the solution itself serves as a polytime certificate, putting it in NP. What else can be said about the complexity of this problem?

Do you mean we can take such a Diophantine problem and encode as an SAT instance? This seems right, but the other direction is the more interesting one and it isn't obvious to me: that any SAT formula can be encoded as such a norm-bounded Diophantine equation.
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R HahnAug 23 '10 at 3:14

no i meant it the correct way. Take a SAT formula and encode it as a polynomial using $x$ for a variable, $1-x$ for its negation, and so on.
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Suresh VenkatAug 23 '10 at 4:45

note also that it's easy to encode the bounded norm constraint as well, since the total sum of all variables is at most $n$, in addition to the integer constraint.
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Suresh VenkatAug 23 '10 at 4:46

right, of course. thank you. If I rephrase the problem in terms of at most 9 unknowns -- which is sufficient so that the unbounded decision problem is undecidable -- this reduction isn't so straightforward. I am editing the question to reflect this more specific case.
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R HahnAug 23 '10 at 5:16